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, Volume 36, Issue 4, pp 285–300 | Cite as

Graph isomorphism and theorems of Birkhoff type

  • G. Tinhofer
Contributed Papers

Abstract

Two graphsG andG′ having adjacency matricesA andB are called ds-isomorphic iff there is a doubly stochastic matrixX satisfyingXA=BX.Ds-isomorphism is a relaxation of the classical isomorphism relation. In section 2 a complete set of invariants with respect tods-isomorphism is given. In the case whereA=B (ds-automorphism) the main question is: For which graphsG the polytope ofds-automorphisms ofG equals the convex hull of the automorphisms ofG? In section 3 a positive answer to this question is given for the cases whereG is a tree or whereG is a cycle. The corresponding theorems are analoga to the well known theorem of Birkhoff on doubly stochastic matrices.

AMS Subject Classifications

05C05 05C25 05C50 15A24 15A51 20B25 20B35 52A25 68E10 

Key words

Graph isomorphism doubly stochastic matrices convex polytops trees cycles 

Isomorphie von Graphen und Theoreme vom Birkhoff-Typ

Zusammenfassung

Zwei GraphenG undG′ werdends-isomorph genannt, wenn eine doppelt stochastische MatrixX existiert mitXA=BX, wobeiA undB die Adjazenzmatrizen vonG undG′ sind.Ds-Isomorphie ist eine Vergröberung der klassischen Isomorphierelation. In Abschnitt 2 wird ein vollständiges Invariantensystem bezüglichds-Isomorphie vorgestellt. Für den FallA=B (ds-Automorphismus) lautet die Hauptfrage: Für welche GraphenG ist das Polytop derds-Automorphismen gleich der konvexen Hülle der klassischen Automorphismen? In Abschnitt 3 wird diese Frage für Kreise und Bäume positiv beantwortet. Die entsprechenden Theoreme sind Analoga zu dem bekannten Satz von Birkhoff über doppelt stochastische Matrizen.

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • G. Tinhofer
    • 1
  1. 1.Institut für MathematikTechnische UniversitätMünchen 2Germany

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